Question

Let S be the set of all values of θ Ε [-π, π] for which the system of linear equations
x+y+√3z=0
-x+(tanθ)y+ √7z=0
x+y+(tanθ)z = 0 has non-trivial solution.
Then 120/π ∑θ_θ∈s is equal to

• A. 10
• B. 20
• C. 30
• D. 40

Answer 1

The Correct Option is B

The determinant is given by:
\[ D = \left| \begin{matrix} 1 & 1 & \sqrt{3} \\ -1 & \tan \theta & \sqrt{7} \\ 1 & 1 & \tan \theta \end{matrix} \right| = 0 \] 

Simplifying the determinant expression: \[ \tan^2 \theta - (\sqrt{3} - 1) - \sqrt{3} = 0 \] 

Solving for \( \tan \theta \): \[ \tan \theta = \sqrt{3}, -1 \] 

Therefore, the possible values for \( \theta \) are: \[ \theta = \left\{ \frac{\pi}{3}, -\frac{2\pi}{3}, -\frac{\pi}{4}, \frac{3\pi}{4} \right\} \] 

Now, calculating the sum of angles: \[ \frac{120}{\pi} \left( \sum \theta \right) = \frac{120}{\pi} \times \frac{\pi}{6} = 20 \quad (\text{Option 2}) \]